According to this [link](http://www.princeton.edu/~sspirkl/openproblems.html) (see point $15$), Matt DeVos and David Wood can prove that every digraph in which each vertex has an out-neighbour has a good $4$-colouring (I am using their terminology).  

**Update.** The paper [Majority Colourings of Digraphs](http://arxiv.org/pdf/1608.03040v1.pdf) by Kreutzer, Oum, Seymour, van der Zypen (the OP), and Wood is now available on the arXiv. They give a short proof for $4$ colours and provide evidence for the conjecture with $3$ colours.